Structure-Function Relationships behind the Phenomenon of Cognitive Resilience in Neurology: Insights for Neuroscience and Medicine
Figure 6
Information transfer and functional resilience in a model of small network subjected to disconnection. The principles, basic nodes, and (two-way) connections of the model are the same as in Figure 5(b). Upper-tier. Graphs of different states of the basic model. Two processes A and B implement some functions within, respectively, nodes (1, 2) and nodes (4, 5). It is assumed that the processes rely on an exchange of information between their respective nodes to compute the output of the functions. Disruption of information flow between nodes after structural disconnection (b, c, d) implies a failure of the process. A central node 3 or “hub” is connected to all the other nodes. Node 3 can be “offline” (a, b); that is, its effective connectivity with the other nodes is very low, and thus signals cannot flow through it even in the presence of structural connections. Alternatively it can be “online” (c, d); that is, its effective connectivity with the other nodes is high, and thus signals can directly flow through it. When node 3 is online, cross-talks and interferences between processes will arise (c), except if the node internal structure fully supports pluripotency (d), in case that information flow between nodes 1 and 2 is restored indirectly via node 3 without interference. Lower-Tier. Adjacency matrices of effective connectivity (bottom row) and associated mutual information (top row) for the different states of the network in (a, b, c, and d). Each square within the matrix represents the strength of the corresponding parameter between two nodes among the 5 nodes of the network. The color bars on the right indicate the relative strength of the parameters. Model specification and simulations. In order to demonstrate the property illustrated in (a, b, c, and d), we used a simple generative model. Binary matrices of structural connectivity were first defined to implement the structure of connectivity. Matrices of effective connectivity were defined by reweighting the matrix of structural connectivity in order to implement the different states of effective connectivity in the network in (a), (b), (c), and (d) (e.g., high versus low connectivity; disconnection). Effective connectivity matrices were used as an input to a spatial autoregressive model (SAR), which demonstrates the best predictive power among many other more complex and realistic models in predicting empirical functional connectivity (from fRMI), based on empirical structural connectivity (from DWI) (see [102]). The SAR model assumes a stationary, multivariate, independently, and identically distributed Gaussian process with covariance matrix . It corresponds to the equilibrium solution of a simple linear dynamical system of nodes interacting through a set of structural connections, with effective connectivity matrix . The covariance matrix has a closed form solution with the SAR model and can be computed based on the matrix of effective connectivity : , with “” being the variance of the process, “” being the identity matrix, and , respectively, being the inverse and transpose of the matrices within the parenthesis, and being a parameter of overall connectivity within the network (here was set to .8). The covariance corresponds to what is measured as functional connectivity in functional neuroimaging: functional connectivity is generally defined as the covariance between measured signals. A correlation matrix can be derived from the covariance matrix , and there is a simple relation between the correlation in the dynamics of two nodes and and their mutual information (expressed in bits of information) because of the statistical properties of the SAR model. Thus , with ln(·) being the natural logarithm and sqrt(·) being the square root functions. (a) Information flows between nodes 1 and 2, thanks to their high effective connectivity; mutual information is high between them. (b) Nodes 1 and 2 have been structurally disconnected, information does not flow, and their mutual information is zero. (c) Nodes 1 and 2 have been structurally disconnected, information does not flow directly, but node 3 has been brought online and information can flow through it indirectly. Mutual information between nodes 1 and 2 is partially restored, but there are also interfering cross-talks between nodes (1, 2) and (4, 5) through node 3 (see green squares in the corresponding matrix). (d) Nodes 1 and 2 have been structurally disconnected, information does not flow directly, and node 3 has been brought online but in a pluripotent manner that filters out cross-talks, resulting in no mutual information between nodes (1, 2) and (4, 5). Mutual information between nodes 1 and 2 is partially restored, as signals can flow through node 3. Information processing has been partially renormalized by playing on effective connectivity over redundancies in the model.
